Guess there're quite a few loose ends i need to tie...
If Michelson and Morlet had used a Sagnac interferometer (one where two rays of light turn around some space in opposite directions) rather than a cross-shaped interferometer, they would have found their "aether"... Or believed they did. They would have seen the sort of shift in the interferences that they were looking for. They would have proven the earth movement with lights and mirrors... But a Sagnac interferometer can only detect rotation, not inertial translation.
Does that mean that rotation is absolute, while translation is not?
First, let see what the earth's rotation is relative to: the sun, the stars, etc. We can agree that either the stars turn around an immobile earth, or our earth itself turns around an axis as a toupee, generating the illusion of a rotating universe. Whatever the case, the rotation is relative to something considered (at least in first approximation) as immobile.
Lay asked at some point why we couldn't consider earth as immobile and the sky turning around it, as was the view before Copernicus. The answer, paraphrasing Poincaré, is that we could, but the laws of sideral motion would be immensely more complicated than in our current non-earth centric system, and much harder to understand intuitively. Galaxies far away will go so very fast, and all that... It wouldn't be elegant nor convenient, but it can be done.
In SR terms, a summary of the above would be: The laws of motion and light and time, as SR expresses them, work when measured through any inertial frame, but get APPARENTLY messed up when the universe is observed using a frame that's affected by non-inertial movements, such as rotation or acceleration. Like the balls on a billiard table riding on a roller-coaster would get all messed up, a frame's own acceleration (in the broad physical sense of the word ie including rotation, vibration, change in direction, etc) introduces apparent perturbations and oddities when reading and plotting movements in that frame.
So in summary, the earth rotates around its north south poles axis when observed in any inertial frame
. Any such frame would allow one to plot the movement of the entire solar system the same way, and would yield the exact same results in terms of orbits, sideral years, precessions, etc.
Does that mean the rotation of the earth is absolute? It depends...
If the pre-copernician situation described above does not bother you, you agree that earth's rotation is relative.
What if it DOES bother you? What if you'd rather think you're not the center of the universe to benefit from much simpler physics? Are you then entitled to conclude that earth's rotation is absolute?
I guess yes, if your definition of absolute is that which, if not true, would result in absurdly and needlessly complicated physics. My definition of "absolute" is different: I think it means "in and by itself, not relative to anything else". Under this definition, the earth doesn't really turn "in and by itself". It turns relative to the rest of the universe.